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#LyX 2.4 created this file. For more info see https://www.lyx.org/
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\begin_document
\begin_header
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\index Index
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\end_header
\begin_body
\begin_layout Section
Symmetries
\begin_inset CommandInset label
LatexCommand label
name "sec:Symmetries"
\end_inset
\end_layout
\begin_layout Standard
If the system has nontrivial point group symmetries, group theory gives
additional understanding of the system properties, and can be used to reduce
the computational costs.
\end_layout
\begin_layout Standard
As an example, if our system has a
\begin_inset Formula $D_{2h}$
\end_inset
symmetry and our truncated
\begin_inset Formula $\left(I-T\trops\right)$
\end_inset
matrix has size
\begin_inset Formula $N\times N$
\end_inset
,
\begin_inset Note Note
status open
\begin_layout Plain Layout
nepoužívám
\begin_inset Formula $N$
\end_inset
už v jiném kontextu?
\end_layout
\end_inset
it can be block-diagonalized into eight blocks of size about
\begin_inset Formula $N/8\times N/8$
\end_inset
, each of which can be LU-factorised separately (this is due to the fact
that
\begin_inset Formula $D_{2h}$
\end_inset
has eight different one-dimensional irreducible representations).
This can reduce both memory and time requirements to solve the scattering
problem
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Multiple-scattering problem block form"
plural "false"
caps "false"
noprefix "false"
\end_inset
by a factor of 64.
\end_layout
\begin_layout Standard
In periodic systems (problems
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Multiple-scattering problem unit cell block form"
plural "false"
caps "false"
noprefix "false"
\end_inset
,
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:lattice mode equation"
plural "false"
caps "false"
noprefix "false"
\end_inset
) due to small number of particles per unit cell, the costliest part is
usually the evaluation of the lattice sums in the
\begin_inset Formula $W\left(\omega,\vect k\right)$
\end_inset
matrix, not the linear algebra.
However, the lattice modes can be searched for in each irrep separately,
and the irrep dimension gives a priori information about mode degeneracy.
\end_layout
\begin_layout Subsection
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Excitation coefficients under point group operations
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\end_layout
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\begin_layout Standard
\begin_inset Note Note
status open
\begin_layout Plain Layout
TODO Zkontrolovat všechny vzorečky zde!!!
\end_layout
\end_inset
In order to use the point group symmetries, we first need to know how they
affect our basis functions, i.e.
the VSWFs.
\end_layout
\begin_layout Standard
Let
\begin_inset Formula $g$
\end_inset
be a member of orthogonal group
\begin_inset Formula $O(3)$
\end_inset
, i.e.
a 3D point rotation or reflection operation that transforms vectors in
\begin_inset Formula $\reals^{3}$
\end_inset
with an orthogonal matrix
\begin_inset Formula $R_{g}$
\end_inset
:
\begin_inset Formula
\[
\vect r\mapsto R_{g}\vect r.
\]
\end_inset
Spherical harmonics
\begin_inset Formula $\ush lm$
\end_inset
, being a basis the
\begin_inset Formula $l$
\end_inset
-dimensional representation of
\begin_inset Formula $O(3)$
\end_inset
, transform as
\begin_inset CommandInset citation
LatexCommand cite
after "???"
key "dresselhaus_group_2008"
literal "false"
\end_inset
\begin_inset Formula
\[
\ush lm\left(R_{g}\uvec r\right)=\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\ush l{m'}\left(\uvec r\right)
\]
\end_inset
where
\begin_inset Formula $D_{m,m'}^{l}\left(g\right)$
\end_inset
denotes the elements of the
\emph on
Wigner matrix
\emph default
representing the operation
\begin_inset Formula $g$
\end_inset
.
By their definition, vector spherical harmonics
\begin_inset Formula $\vsh 2lm,\vsh 3lm$
\end_inset
transform in the same way,
\begin_inset Formula
\begin{align*}
\vsh 2lm\left(R_{g}\uvec r\right) & =\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\vsh 2l{m'}\left(\uvec r\right),\\
\vsh 3lm\left(R_{g}\uvec r\right) & =\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\vsh 3l{m'}\left(\uvec r\right),
\end{align*}
\end_inset
but the remaining set
\begin_inset Formula $\vsh 1lm$
\end_inset
transforms differently due to their pseudovector nature stemming from the
cross product in their definition:
\begin_inset Formula
\[
\vsh 3lm\left(R_{g}\uvec r\right)=\sum_{m'=-l}^{l}\widetilde{D_{m,m'}^{l}}\left(g\right)\vsh 3l{m'}\left(\uvec r\right),
\]
\end_inset
where
\begin_inset Formula $\widetilde{D_{m,m'}^{l}}\left(g\right)=D_{m,m'}^{l}\left(g\right)$
\end_inset
if
\begin_inset Formula $g$
\end_inset
is a proper rotation, but for spatial inversion operation
\begin_inset Formula $i:\vect r\mapsto-\vect r$
\end_inset
we have
\begin_inset Formula $\widetilde{D_{m,m'}^{l}}\left(i\right)=\left(-1\right)^{l+m}D_{m,m'}^{l}\left(i\right)$
\end_inset
.
The transformation behaviour of vector spherical harmonics directly propagates
to the spherical vector waves, cf.
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:VSWF regular"
plural "false"
caps "false"
noprefix "false"
\end_inset
,
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:VSWF outgoing"
plural "false"
caps "false"
noprefix "false"
\end_inset
:
\begin_inset Formula
\begin{align*}
\vswfouttlm 1lm\left(R_{g}\vect r\right) & =\sum_{m'=-l}^{l}\widetilde{D_{m,m'}^{l}}\left(g\right)\vswfouttlm 1l{m'}\left(\vect r\right),\\
\vswfouttlm 2lm\left(R_{g}\vect r\right) & =\sum_{m'=-l}^{l}D_{m,m'}^{l}\left(g\right)\vswfouttlm 2l{m'}\left(\vect r\right),
\end{align*}
\end_inset
(and analogously for the regular waves
\begin_inset Formula $\vswfrtlm{\tau}lm$
\end_inset
).
\begin_inset Note Note
status open
\begin_layout Plain Layout
TODO víc obdivu.
\end_layout
\end_inset
For convenience, we introduce the symbol
\begin_inset Formula $D_{m,m'}^{\tau l}$
\end_inset
that describes the transformation of both types (
\begin_inset Quotes eld
\end_inset
magnetic
\begin_inset Quotes erd
\end_inset
and
\begin_inset Quotes eld
\end_inset
electric
\begin_inset Quotes erd
\end_inset
) of waves at once:
\begin_inset Formula
\[
\vswfouttlm{\tau}lm\left(R_{g}\vect r\right)=\sum_{m'=-l}^{l}D_{m,m'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(\vect r\right).
\]
\end_inset
Using these, we can express the VSWF expansion
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:E field expansion"
plural "false"
caps "false"
noprefix "false"
\end_inset
of the electric field around origin in a rotated/reflected system,
\begin_inset Formula
\[
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\vect E\left(\omega,R_{g}\vect r\right)=\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoefftlm{\tau}lmD_{m,m'}^{\tau l}\left(g\right)\vswfrtlm{\tau}l{m'}\left(k\vect r\right)+\outcoefftlm{\tau}lmD_{m,m'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(k\vect r\right)\right),
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\]
\end_inset
which, together with the
\begin_inset Formula $T$
\end_inset
-matrix definition,
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:T-matrix definition"
plural "false"
caps "false"
noprefix "false"
\end_inset
can be used to obtain a
\begin_inset Formula $T$
\end_inset
-matrix of a rotated or mirror-reflected particle.
Let
\begin_inset Formula $T$
\end_inset
be the
\begin_inset Formula $T$
\end_inset
-matrix of an original particle; the
\begin_inset Formula $T$
\end_inset
-matrix of a particle physically transformed by operation
\begin_inset Formula $g\in O(3)$
\end_inset
is then
\begin_inset Note Note
status open
\begin_layout Plain Layout
check sides
\end_layout
\end_inset
\begin_inset Formula
\begin{equation}
T'_{\tau lm;\tau'l'm'}=\sum_{\mu=-l}^{l}\sum_{\mu'=-l'}^{l'}\left(D_{\mu,m}^{\tau l}\left(g\right)\right)^{*}T_{\tau l\mu;\tau'l'm'}D_{m',\mu'}^{\tau l}\left(g\right).\label{eq:T-matrix of a transformed particle}
\end{equation}
\end_inset
If the particle is symmetric (so that
\begin_inset Formula $g$
\end_inset
produces a particle indistinguishable from the original one), the
\begin_inset Formula $T$
\end_inset
-matrix must remain invariant under the transformation
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:T-matrix of a transformed particle"
plural "false"
caps "false"
noprefix "false"
\end_inset
,
\begin_inset Formula $T'_{\tau lm;\tau'l'm'}=T{}_{\tau lm;\tau'l'm'}$
\end_inset
.
Explicit forms of these invariance properties for the most imporant point
group symmetries can be found in
\begin_inset CommandInset citation
LatexCommand cite
key "schulz_point-group_1999"
literal "false"
\end_inset
.
\end_layout
\begin_layout Standard
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If the field expansion is done around a point
\begin_inset Formula $\vect r_{p}$
\end_inset
different from the global origin, as in
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:E field expansion multiparticle"
plural "false"
caps "false"
noprefix "false"
\end_inset
, we have
\begin_inset Formula
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\begin{multline}
\vect E\left(\omega,R_{g}\vect r\right)=\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoeffptlm p{\tau}lmD_{m',\mu'}^{\tau l}\left(g\right)\vswfrtlm{\tau}l{m'}\left(k\left(\vect r-R_{g}\vect r_{p}\right)\right)\right.+\\
+\left.\outcoeffptlm p{\tau}lmD_{m',\mu'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(k\left(\vect r-R_{g}\vect r_{p}\right)\right)\right).\label{eq:rotated E field expansion around outside origin}
\end{multline}
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\end_inset
\end_layout
\begin_layout Standard
\begin_inset Float figure
placement document
alignment document
wide false
sideways false
status open
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\begin_layout Plain Layout
\align center
\begin_inset CommandInset include
LatexCommand input
filename "orbits.tex"
literal "true"
\end_inset
\end_layout
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\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
Scatterer orbits under
\begin_inset Formula $D_{2}$
\end_inset
symmetry.
Particles
\begin_inset Formula $A,B,C,D$
\end_inset
lie outside of origin or any mirror planes, and together constitute an
orbit of the size equal to the order of the group,
\begin_inset Formula $\left|D_{2}\right|=4$
\end_inset
.
Particles
\begin_inset Formula $E,F$
\end_inset
lie on the
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\begin_inset Formula $yz$
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\end_inset
plane, hence the corresponding reflection maps each of them to itself,
but the
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\begin_inset Formula $xz$
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\end_inset
reflection (or the
\begin_inset Formula $\pi$
\end_inset
rotation around the
\begin_inset Formula $z$
\end_inset
axis) maps them to each other, forming a particle orbit of size 2
\begin_inset Note Note
status open
\begin_layout Plain Layout
=???
\end_layout
\end_inset
.
The particle
\begin_inset Formula $O$
\end_inset
in the very origin is always mapped to itself, constituting its own orbit.
\begin_inset CommandInset label
LatexCommand label
name "fig:D2-symmetric structure particle orbits"
\end_inset
\end_layout
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Note Note
status open
\begin_layout Plain Layout
TODO restructure this
\end_layout
\end_inset
With these transformation properties in hand, we can proceed to the effects
of point symmetries on the whole many-particle system.
Let us have a many-particle system symmetric with respect to a point group
\begin_inset Formula $G$
\end_inset
.
A symmetry operation
\begin_inset Formula $g\in G$
\end_inset
determines a permutation of the particles:
\begin_inset Formula $p\mapsto\pi_{g}(p)$
\end_inset
,
\begin_inset Formula $p\in\mathcal{P}$
\end_inset
.
For a given particle
\begin_inset Formula $p$
\end_inset
, we will call the set of particles onto which any of the symmetries maps
the particle
\begin_inset Formula $p$
\end_inset
, i.e.
the set
\begin_inset Formula $\left\{ \pi_{g}\left(p\right);g\in G\right\} $
\end_inset
, as the
\emph on
orbit
\emph default
of particle
\begin_inset Formula $p$
\end_inset
.
The whole set
\begin_inset Formula $\mathcal{P}$
\end_inset
can therefore be divided into the different particle orbits; an example
is in Fig.
\begin_inset CommandInset ref
LatexCommand ref
reference "fig:D2-symmetric structure particle orbits"
plural "false"
caps "false"
noprefix "false"
\end_inset
.
The importance of the particle orbits stems from the following: in the
multiple-scattering problem, outside of the scatterers
\begin_inset Note Note
status open
\begin_layout Plain Layout
< FIXME
\end_layout
\end_inset
one has
\begin_inset Formula
\begin{align}
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\vect E\left(\omega,R_{g}\vect r\right) & =\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoeffptlm p{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfrtlm{\tau}l{m'}\left(k\left(\vect r-\vect r_{\pi_{g}(p)}\right)\right)\right.+\label{eq:rotated E field expansion around outside origin-1}\\
& \quad+\left.\outcoeffptlm p{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(k\left(\vect r-\vect r_{p}\right)\right)\right)\\
& =\sum_{\tau=1,2}\sum_{l=1}^{\infty}\sum_{m=-l}^{+l}\sum_{m'=-l}^{l}\left(\rcoeffptlm{\pi_{g}^{-1}(p)}{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfrtlm{\tau}l{m'}\left(k\left(\vect r-\vect r_{p}\right)\right)\right.+\\
& \quad+\left.\outcoeffptlm{\pi_{g}^{-1}(p)}{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right)\vswfouttlm{\tau}l{m'}\left(k\left(\vect r-\vect r_{p}\right)\right)\right).
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\end{align}
\end_inset
This means that the field expansion coefficients
\begin_inset Formula $\rcoeffp p,\outcoeffp p$
\end_inset
transform as
\begin_inset Formula
\begin{align}
\rcoeffptlm p{\tau}lm & \mapsto\rcoeffptlm{\pi_{g}^{-1}(p)}{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right),\nonumber \\
\outcoeffptlm p{\tau}lm & \mapsto\outcoeffptlm{\pi_{g}^{-1}(p)}{\tau}lmD_{m,\mu'}^{\tau l}\left(g\right).\label{eq:excitation coefficient under symmetry operation}
\end{align}
\end_inset
Obviously, the expansion coefficients belonging to particles in different
orbits do not mix together.
As before, we introduce a short-hand block-matrix notation for
\begin_inset CommandInset ref
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LatexCommand eqref
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reference "eq:excitation coefficient under symmetry operation"
plural "false"
caps "false"
noprefix "false"
\end_inset
(TODO avoid notation clash here in a more consistent and readable way!)
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{align}
\rcoeff & \mapsto J\left(g\right)a,\nonumber \\
\outcoeff & \mapsto J\left(g\right)\outcoeff.\label{eq:excitation coefficient under symmetry operation block form}
\end{align}
\end_inset
\begin_inset Note Note
status open
\begin_layout Plain Layout
The matrices
\begin_inset Formula $D\left(g\right)$
\end_inset
,
\begin_inset Formula $g\in G$
\end_inset
will play a crucial role blablabla
\end_layout
\end_inset
If the particle indices are ordered in a way that the particles belonging
to the same orbit are grouped together,
\begin_inset Formula $J\left(g\right)$
\end_inset
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will be a block-diagonal unitary matrix, each block (also unitary) representing
the action of
\begin_inset Formula $g$
\end_inset
on one particle orbit.
All the
\begin_inset Formula $J\left(g\right)$
\end_inset
s make together a (reducible) linear representation of
\begin_inset Formula $G$
\end_inset
.
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\end_layout
\begin_layout Subsection
Irrep decomposition
\end_layout
\begin_layout Standard
Knowledge of symmetry group actions
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\begin_inset Formula $J\left(g\right)$
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\end_inset
on the field expansion coefficients give us the possibility to construct
a symmetry adapted basis in which we can block-diagonalise the multiple-scatter
ing problem matrix
\begin_inset Formula $\left(I-TS\right)$
\end_inset
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.
Let
\begin_inset Formula $\Gamma_{n}$
\end_inset
be the
\begin_inset Formula $d_{n}$
\end_inset
-dimensional irreducible matrix representations of
\begin_inset Formula $G$
\end_inset
consisting of matrices
\begin_inset Formula $D^{\Gamma_{n}}\left(g\right)$
\end_inset
.
Then the projection operators
\begin_inset Formula
\[
P_{kl}^{\left(\Gamma_{n}\right)}\equiv\frac{d_{n}}{\left|G\right|}\sum_{g\in G}\left(D^{\Gamma_{n}}\left(g\right)\right)_{kl}^{*}J\left(g\right),\quad k,l=1,\dots,d_{n}
\]
\end_inset
project the full scattering system field expansion coefficient vectors
\begin_inset Formula $\rcoeff,\outcoeff$
\end_inset
onto a subspace corresponding to the irreducible representation
\begin_inset Formula $\Gamma_{n}$
\end_inset
.
The projectors can be used to construct a unitary transformation
\begin_inset Formula $U$
\end_inset
with components
\begin_inset Formula
\begin{equation}
U_{nri;p\tau lm}=\frac{d_{n}}{\left|G\right|}\sum_{g\in G}\left(D^{\Gamma_{n}}\left(g\right)\right)_{rr}^{*}J\left(g\right)_{p'\tau'l'm'(nri);p\tau lm}\label{eq:SAB unitary transformation operator}
\end{equation}
\end_inset
where
\begin_inset Formula $r$
\end_inset
goes from
\begin_inset Formula $1$
\end_inset
through
\begin_inset Formula $d_{n}$
\end_inset
and
\begin_inset Formula $i$
\end_inset
goes from 1 through the multiplicity of irreducible representation
\begin_inset Formula $\Gamma_{n}$
\end_inset
in the (reducible) representation of
\begin_inset Formula $G$
\end_inset
spanned by the field expansion coefficients
\begin_inset Formula $\rcoeff$
\end_inset
or
\begin_inset Formula $\outcoeff$
\end_inset
.
The indices
\begin_inset Formula $p',\tau',l',m'$
\end_inset
are given by an arbitrary bijective mapping
\begin_inset Formula $\left(n,r,i\right)\mapsto\left(p',\tau',l',m'\right)$
\end_inset
with the constraint that for given
\begin_inset Formula $n,r,i$
\end_inset
there are at least some non-zero elements
\begin_inset Formula $U_{nri;p\tau lm}$
\end_inset
.
For details, we refer the reader to textbooks about group representation
theory
\begin_inset Note Note
status open
\begin_layout Plain Layout
or linear representations?
\end_layout
\end_inset
, e.g.
\begin_inset CommandInset citation
LatexCommand cite
after "Chapter 4"
key "dresselhaus_group_2008"
literal "false"
\end_inset
or
\begin_inset CommandInset citation
LatexCommand cite
after "???"
key "bradley_mathematical_1972"
literal "false"
\end_inset
.
The transformation given by
\begin_inset Formula $U$
\end_inset
transforms the excitation coefficient vectors
\begin_inset Formula $\rcoeff,\outcoeff$
\end_inset
into a new,
\emph on
symmetry-adapted basis
\emph default
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.
\end_layout
\begin_layout Standard
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One can show that if an operator
\begin_inset Formula $M$
\end_inset
acting on the excitation coefficient vectors is invariant under the operations
of group
\begin_inset Formula $G$
\end_inset
, meaning that
\begin_inset Formula
\[
\forall g\in G:J\left(g\right)MJ\left(g\right)^{\dagger}=M,
\]
\end_inset
then in the symmetry-adapted basis,
\begin_inset Formula $M$
\end_inset
is block diagonal, or more specifically
\begin_inset Formula
\[
M_{\Gamma,r,i;\Gamma',r',j}^{\mathrm{s.a.b.}}=\frac{\delta_{\Gamma\Gamma'}\delta_{ij}}{d_{\Gamma}}\sum_{q}M{}_{\Gamma,r,q;\Gamma',r',q}^{\mathrm{s.a.b.}}.
\]
\end_inset
Both the
\begin_inset Formula $T$
\end_inset
and
\begin_inset Formula $\trops$
\end_inset
operators (and trivially also the identity
\begin_inset Formula $I$
\end_inset
) in
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Multiple-scattering problem block form"
plural "false"
caps "false"
noprefix "false"
\end_inset
are invariant under the actions of whole system symmetry group, so
\begin_inset Formula $\left(I-T\trops\right)$
\end_inset
is also invariant, hence
\begin_inset Formula $U\left(I-T\trops\right)U^{\dagger}$
\end_inset
is a block-diagonal matrix, and the problem
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Multiple-scattering problem block form"
plural "false"
caps "false"
noprefix "false"
\end_inset
can be solved for each block separately.
\end_layout
\begin_layout Standard
From the computational perspective, it is important to note that
\begin_inset Formula $U$
\end_inset
is at least as sparse as
\begin_inset Formula $J\left(g\right)$
\end_inset
(which is
\begin_inset Quotes eld
\end_inset
orbit-block
\begin_inset Quotes erd
\end_inset
diagonal), hence the block-diagonalisation can be performed fast.
\begin_inset Note Note
status open
\begin_layout Plain Layout
Kvantifikovat!
\end_layout
\end_inset
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\end_layout
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\begin_layout Subsection
Periodic systems
\end_layout
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\begin_layout Standard
For periodic systems, we can in similar manner also block-diagonalise the
\begin_inset Formula $M\left(\omega,\vect k\right)=\left(I-W\left(\omega,\vect k\right)T\left(\omega\right)\right)$
\end_inset
from the left hand side of eqs.
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:Multiple-scattering problem unit cell block form"
plural "false"
caps "false"
noprefix "false"
\end_inset
,
\begin_inset CommandInset ref
LatexCommand eqref
reference "eq:lattice mode equation"
plural "false"
caps "false"
noprefix "false"
\end_inset
.
Hovewer, in this case,
\begin_inset Formula $W\left(\omega,\vect k\right)$
\end_inset
is in general not invariant under the whole point group symmetry subgroup
of the system geometry due to the
\begin_inset Formula $\vect k$
\end_inset
dependence.
In other words, only those point symmetries that the
\begin_inset Formula $e^{i\vect k\cdot\vect r}$
\end_inset
modulation does not break are preserved, and no preservation of point symmetrie
s happens unless
\begin_inset Formula $\vect k$
\end_inset
lies somewhere in the high-symmetry parts of the Brillouin zone.
However, the high-symmetry points are usually the ones of the highest physical
interest, for it is where the band edges
\begin_inset Note Note
status open
\begin_layout Plain Layout
or
\begin_inset Quotes eld
\end_inset
dirac points
\begin_inset Quotes erd
\end_inset
\end_layout
\end_inset
are typically located.
\end_layout
\begin_layout Standard
The transformation to the symmetry adapted basis
\begin_inset Formula $U$
\end_inset
is constructed in a similar way as in the finite case, but because we do
not work with all the (infinite number of) scatterers but only with one
unit cell, additional phase factors
\begin_inset Formula $e^{i\vect k\cdot\vect r_{p}}$
\end_inset
appear in the per-unit-cell group action
\begin_inset Formula $J(g)$
\end_inset
.
This is illustrated in Fig.
\begin_inset CommandInset ref
LatexCommand ref
reference "Phase factor illustration"
plural "false"
caps "false"
noprefix "false"
\end_inset
.
\begin_inset Float figure
placement document
alignment document
wide false
sideways false
status open
\begin_layout Plain Layout
\end_layout
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
\begin_inset CommandInset label
LatexCommand label
name "Phase factor illustration"
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Plain Layout
\end_layout
\end_inset
\end_layout
\begin_layout Standard
More rigorous analysis can be found e.g.
in
\begin_inset CommandInset citation
LatexCommand cite
after "chapters 10–
key "dresselhaus_group_2008"
literal "true"
\end_inset
.
\end_layout
\begin_layout Standard
\begin_inset Note Note
status open
\begin_layout Plain Layout
In the group-theoretical terminology, blablabla little groups blabla bla...
\end_layout
\end_inset
\end_layout
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\begin_layout Standard
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\begin_inset Note Note
status open
\begin_layout Plain Layout
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A general overview of utilizing group theory to find lattice modes at high-symme
try points of the Brillouin zone can be found e.g.
in
\begin_inset CommandInset citation
LatexCommand cite
after "chapters 10–
key "dresselhaus_group_2008"
literal "true"
\end_inset
; here we use the same notation.
\end_layout
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\begin_layout Plain Layout
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We analyse the symmetries of the system in the same VSWF representation
as used in the
\begin_inset Formula $T$
\end_inset
-matrix formalism introduced above.
We are interested in the modes at the
\begin_inset Formula $\Kp$
\end_inset
-point of the hexagonal lattice, which has the
\begin_inset Formula $D_{3h}$
\end_inset
point symmetry.
The six irreducible representations (irreps) of the
\begin_inset Formula $D_{3h}$
\end_inset
group are known and are available in the literature in their explicit forms.
In order to find and classify the modes, we need to find a decomposition
of the lattice mode representation
\begin_inset Formula $\Gamma_{\mathrm{lat.mod.}}=\Gamma^{\mathrm{equiv.}}\otimes\Gamma_{\mathrm{vec.}}$
\end_inset
into the irreps of
\begin_inset Formula $D_{3h}$
\end_inset
.
The equivalence representation
\begin_inset Formula $\Gamma^{\mathrm{equiv.}}$
\end_inset
is the
\begin_inset Formula $E'$
\end_inset
representation as can be deduced from
\begin_inset CommandInset citation
LatexCommand cite
after "eq. (11.19)"
key "dresselhaus_group_2008"
literal "true"
\end_inset
, eq.
(11.19) and the character table for
\begin_inset Formula $D_{3h}$
\end_inset
.
\begin_inset Formula $\Gamma_{\mathrm{vec.}}$
\end_inset
operates on a space spanned by the VSWFs around each nanoparticle in the
unit cell (the effects of point group operations on VSWFs are described
in
\begin_inset CommandInset citation
LatexCommand cite
key "schulz_point-group_1999"
literal "true"
\end_inset
).
This space can be then decomposed into invariant subspaces of the
\begin_inset Formula $D_{3h}$
\end_inset
using the projectors
\begin_inset Formula $\hat{P}_{ab}^{\left(\Gamma\right)}$
\end_inset
defined by
\begin_inset CommandInset citation
LatexCommand cite
after "eq. (4.28)"
key "dresselhaus_group_2008"
literal "true"
\end_inset
.
This way, we obtain a symmetry adapted basis
\begin_inset Formula $\left\{ \vect b_{\Gamma,r,i}^{\mathrm{s.a.b.}}\right\} $
\end_inset
as linear combinations of VSWFs
\begin_inset Formula $\vswfs lm{p,t}$
\end_inset
around the constituting nanoparticles (labeled
\begin_inset Formula $p$
\end_inset
),
\begin_inset Formula
\[
\vect b_{\Gamma,r,i}^{\mathrm{s.a.b.}}=\sum_{l,m,p,t}U_{\Gamma,r,i}^{p,t,l,m}\vswfs lm{p,t},
\]
\end_inset
where
\begin_inset Formula $\Gamma$
\end_inset
stands for one of the six different irreps of
\begin_inset Formula $D_{3h}$
\end_inset
,
\begin_inset Formula $r$
\end_inset
labels the different realisations of the same irrep, and the last index
\begin_inset Formula $i$
\end_inset
going from 1 to
\begin_inset Formula $d_{\Gamma}$
\end_inset
(the dimensionality of
\begin_inset Formula $\Gamma$
\end_inset
) labels the different partners of the same given irrep.
The number of how many times is each irrep contained in
\begin_inset Formula $\Gamma_{\mathrm{lat.mod.}}$
\end_inset
(i.e.
the range of index
\begin_inset Formula $r$
\end_inset
for given
\begin_inset Formula $\Gamma$
\end_inset
) depends on the multipole degree cutoff
\begin_inset Formula $l_{\mathrm{max}}$
\end_inset
.
\end_layout
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\begin_layout Plain Layout
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Each mode at the
\begin_inset Formula $\Kp$
\end_inset
-point shall lie in the irreducible spaces of only one of the six possible
irreps and it can be shown via
\begin_inset CommandInset citation
LatexCommand cite
after "eq. (2.51)"
key "dresselhaus_group_2008"
literal "true"
\end_inset
that, at the
\begin_inset Formula $\Kp$
\end_inset
-point, the matrix
\begin_inset Formula $M\left(\omega,\vect k\right)$
\end_inset
defined above takes a block-diagonal form in the symmetry-adapted basis,
\begin_inset Formula
\[
M\left(\omega,\vect K\right)_{\Gamma,r,i;\Gamma',r',j}^{\mathrm{s.a.b.}}=\frac{\delta_{\Gamma\Gamma'}\delta_{ij}}{d_{\Gamma}}\sum_{q}M\left(\omega,\vect K\right)_{\Gamma,r,q;\Gamma',r',q}^{\mathrm{s.a.b.}}.
\]
\end_inset
This enables us to decompose the matrix according to the irreps and to solve
the singular value problem in each irrep separately, as done in Fig.
\begin_inset CommandInset ref
LatexCommand ref
reference "smfig:dispersions"
\end_inset
(a).
\end_layout
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\end_inset
\end_layout
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\end_body
\end_document
